Question

The function $f:[2,\infty )\to Y$ defined by $f\left(x\right)=x^2-4x+5$ is both one-one & onto if:

• A. $Y=R$
• B. $Y=$$[1,\infty )$
• C. $Y=$$[4,\infty )$
• D. $Y=$$[5,\infty )$

Answer 1

The correct option is B $Y=$$[1,\infty )$

Now we have $f\left(x\right)=x^2-4x+5$, then $f\left(2\right)=1$.

So $f\left(x\right)$ to be a function the range may be either $R$ of $[1,\infty )$ as $1$ belongs to the both of the intervals.

Now, given $f$ is one-one and onto.

Let $y$ be any element in the co-domain of the function such that $f\left(x\right)=y$.

or, $x^2-4x+5=y$

or, $x^2-4x+(5-y)=0$

From this equation we require $x$ to be real.

Then $4^2-4(5-y)\ge 0$

or, $4-(5-y)\ge 0$

or, $y\ge 1$.

So to $x$ be a real number we must have $y\in [1,\infty )$.

The co-domain of the function $f\left(x\right)$ is $[1,\infty )$.

Since the function $f\left(x\right)$ is one-one and onto, so the co-domain of the function is equal to the range of the function.

So $Y=[1,\infty )$.